Journal of Mathematical Psychology 47 (2003) 90–100
Tutorial
Tutorial on maximum likelihood estimation
In Jae Myung*
Department of Psychology, Ohio State University, 1885 Neil Avenue Mall, Columbus, OH 43210-1222, USA
Received 30 November 2001; revised 16 October 2002
Abstract
In this paper, I provide a tutorial exposition on maximum likelihood estimation (MLE). The intended audience of this tutorial are
researchers who practice mathematical modeling of cognition but are unfamiliar with the estimation method. Unlike least-squares
estimation which is primarily a descriptive tool, MLE is a preferred method of parameter estimation in statistics and is an
indispensable tool for many statistical modeling techniques, in particular in non-linear modeling with non-normal data. The purpose
of this paper is to provide a good conceptual explanation of the method with illustrative examples so the reader can have a grasp of
some of the basic principles.
r 2003 Elsevier Science (USA). All rights reserved.
1. Introduction
In psychological science, we seek to uncover general
laws and principles that govern the behavior under
investigation. As these laws and principles are not
directly observable, they are formulated in terms of
hypotheses. In mathematical modeling, such hypotheses about the structure and inner working of the
behavioral process of interest are stated in terms of
parametric families of probability distributions called
models. The goal of modeling is to deduce the form of
the underlying process by testing the viability of such
models.
Once a model is specified with its parameters, and
data have been collected, one is in a position to evaluate
its goodness of fit, that is, how well it fits the observed
data. Goodness of fit is assessed by finding parameter
values of a model that best fits the data—a procedure
called parameter estimation.
There are two general methods of parameter estimation. They are least-squares estimation (LSE) and
maximum likelihood estimation (MLE). The former
has been a popular choice of model fitting in psychology
(e.g., Rubin, Hinton, & Wenzel, 1999; Lamberts, 2000
but see Usher & McClelland, 2001) and is tied to many
familiar statistical concepts such as linear regression,
sum of squares error, proportion variance accounted for
*Fax: +614-292-5601.
E-mail address: myung.1@osu.edu.
(i.e. r2 ), and root mean squared deviation. LSE, which
unlike MLE requires no or minimal distributional
assumptions, is useful for obtaining a descriptive
measure for the purpose of summarizing observed data,
but it has no basis for testing hypotheses or constructing
confidence intervals.
On the other hand, MLE is not as widely recognized
among modelers in psychology, but it is a standard
approach to parameter estimation and inference in
statistics. MLE has many optimal properties in estimation: sufficiency (complete information about the parameter of interest contained in its MLE estimator);
consistency (true parameter value that generated the
data recovered asymptotically, i.e. for data of sufficiently large samples); efficiency (lowest-possible variance of parameter estimates achieved asymptotically);
and parameterization invariance (same MLE solution
obtained independent of the parametrization used). In
contrast, no such things can be said about LSE. As such,
most statisticians would not view LSE as a general
method for parameter estimation, but rather as an
approach that is primarily used with linear regression
models. Further, many of the inference methods in
statistics are developed based on MLE. For example,
MLE is a prerequisite for the chi-square test, the Gsquare test, Bayesian methods, inference with missing
data, modeling of random effects, and many model
selection criteria such as the Akaike information
criterion (Akaike, 1973) and the Bayesian information
criteria (Schwarz, 1978).
0022-2496/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0022-2496(02)00028-7
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
In this tutorial paper, I introduce the maximum
likelihood estimation method for mathematical modeling. The paper is written for researchers who are
primarily involved in empirical work and publish in
experimental journals (e.g. Journal of Experimental
Psychology) but do modeling. The paper is intended to
serve as a stepping stone for the modeler to move
beyond the current practice of using LSE to more
informed modeling analyses, thereby expanding his or
her repertoire of statistical instruments, especially in
non-linear modeling. The purpose of the paper is to
provide a good conceptual understanding of the method
with concrete examples. For in-depth, technically more
rigorous treatment of the topic, the reader is directed to
other sources (e.g., Bickel & Doksum, 1977, Chap. 3;
Casella & Berger, 2002, Chap. 7; DeGroot & Schervish,
2002, Chap. 6; Spanos, 1999, Chap. 13).
91
model’s parameter. As the parameter changes in value,
different probability distributions are generated. Formally, a model is defined as the family of probability
distributions indexed by the model’s parameters.
Let f ðyjwÞ denote the probability density function
(PDF) that specifies the probability of observing data
vector y given the parameter w: Throughout this paper
we will use a plain letter for a vector (e.g. y) and a letter
with a subscript for a vector element (e.g. yi ). The
parameter w ¼ ðw1 ; y; wk Þ is a vector defined on a
multi-dimensional parameter space. If individual observations, yi ’s, are statistically independent of one
another, then according to the theory of probability, the
PDF for the data y ¼ ðy1 ; y; ym Þ given the parameter
vector w can be expressed as a multiplication of PDFs
for individual observations,
f ðy ¼ ðy1 ; y2 ; y; yn Þ j wÞ ¼ f1 ðy1 j wÞ f2 ðy2 j wÞ
?fn ðym j wÞ:
2. Model specification
2.1. Probability density function
From a statistical standpoint, the data vector y ¼
ðy1 ; y; ym Þ is a random sample from an unknown
population. The goal of data analysis is to identify the
population that is most likely to have generated the
sample. In statistics, each population is identified by a
corresponding probability distribution. Associated with
each probability distribution is a unique value of the
ð1Þ
To illustrate the idea of a PDF, consider the simplest
case with one observation and one parameter, that is,
m ¼ k ¼ 1: Suppose that the data y represents the
number of successes in a sequence of 10 Bernoulli trials
(e.g. tossing a coin 10 times) and that the probability of
a success on any one trial, represented by the parameter
w; is 0.2. The PDF in this case is given by
f ðy j n ¼ 10; w ¼ 0:2Þ ¼
10!
ð0:2Þy ð0:8Þ10y
y!ð10 yÞ!
ðy ¼ 0; 1; y; 10Þ
Fig. 1. Binomial probability distributions of sample size n ¼ 10 and probability parameter w ¼ 0:2 (top) and w ¼ 0:7 (bottom).
ð2Þ
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
92
which is known as the binomial distribution with
parameters n ¼ 10; w ¼ 0:2: Note that the number of
trials ðnÞ is considered as a parameter. The shape of this
PDF is shown in the top panel of Fig. 1. If the
parameter value is changed to say w ¼ 0:7; a new PDF
is obtained as
10!
f ðy j n ¼ 10; w ¼ 0:7Þ ¼
ð0:7Þy ð0:3Þ10y
y!ð10 yÞ!
ðy ¼ 0; 1; y; 10Þ
ð3Þ
whose shape is shown in the bottom panel of Fig. 1. The
following is the general expression of the PDF of the
binomial distribution for arbitrary values of w and n:
n!
wy ð1 wÞny
f ðyjn; wÞ ¼
y!ðn yÞ!
ð0pwp1; y ¼ 0; 1; y; nÞ
ð4Þ
which as a function of y specifies the probability of data
y for a given value of n and w: The collection of all such
PDFs generated by varying the parameter across its
range (0–1 in this case for w; nX1) defines a model.
2.2. Likelihood function
Given a set of parameter values, the corresponding
PDF will show that some data are more probable than
other data. In the previous example, the PDF with w ¼
0:2; y ¼ 2 is more likely to occur than y ¼ 5 (0.302 vs.
0.026). In reality, however, we have already observed the
data. Accordingly, we are faced with an inverse
problem: Given the observed data and a model of
interest, find the one PDF, among all the probability
densities that the model prescribes, that is most likely to
have produced the data. To solve this inverse problem,
we define the likelihood function by reversing the roles of
the data vector y and the parameter vector w in f ðyjwÞ;
i.e.
LðwjyÞ ¼ f ðyjwÞ:
ð5Þ
Thus LðwjyÞ represents the likelihood of the parameter
w given the observed data y; and as such is a function of
w: For the one-parameter binomial example in Eq. (4),
the likelihood function for y ¼ 7 and n ¼ 10 is given by
Lðw j n ¼ 10; y ¼ 7Þ ¼ f ðy ¼ 7 j n ¼ 10; wÞ
10! 7
¼
w ð1 wÞ3 ð0pwp1Þ:
7!3!
ð6Þ
The shape of this likelihood function is shown in Fig. 2.
There exist an important difference between the PDF
f ðyjwÞ and the likelihood function LðwjyÞ: As illustrated
in Figs. 1 and 2, the two functions are defined on
different axes, and therefore are not directly comparable
to each other. Specifically, the PDF in Fig. 1 is a
function of the data given a particular set of parameter
values, defined on the data scale. On the other hand, the
likelihood function is a function of the parameter given
a particular set of observed data, defined on the
parameter scale. In short, Fig. 1 tells us the probability
of a particular data value for a fixed parameter, whereas
Fig. 2 tells us the likelihood (‘‘unnormalized probability’’) of a particular parameter value for a fixed data set.
Note that the likelihood function in this figure is a curve
Fig. 2. The likelihood function given observed data y ¼ 7 and sample size n ¼ 10 for the one-parameter model described in the text.
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
because there is only one parameter beside n; which is
assumed to be known. If the model has two parameters,
the likelihood function will be a surface sitting above the
parameter space. In general, for a model with k
parameters, the likelihood function LðwjyÞ takes the
shape of a k-dim geometrical ‘‘surface’’ sitting above a
k-dim hyperplane spanned by the parameter vector w ¼
ðw1 ; y; wk Þ:
3. Maximum likelihood estimation
Once data have been collected and the likelihood
function of a model given the data is determined, one is
in a position to make statistical inferences about the
population, that is, the probability distribution that
underlies the data. Given that different parameter values
index different probability distributions (Fig. 1), we are
interested in finding the parameter value that corresponds to the desired probability distribution.
The principle of maximum likelihood estimation
(MLE), originally developed by R.A. Fisher in the
1920s, states that the desired probability distribution is
the one that makes the observed data ‘‘most likely,’’
which means that one must seek the value of the
parameter vector that maximizes the likelihood function
LðwjyÞ: The resulting parameter vector, which is sought
by searching the multi-dimensional parameter space, is
called the MLE estimate, and is denoted by wMLE ¼
ðw1;MLE ; y; wk;MLE Þ: For example, in Fig. 2, the MLE
estimate is wMLE ¼ 0:7 for which the maximized likelihood value is LðwMLE ¼ 0:7jn ¼ 10; y ¼ 7Þ ¼ 0:267:
The probability distribution corresponding to this
MLE estimate is shown in the bottom panel of Fig. 1.
According to the MLE principle, this is the population
that is most likely to have generated the observed data
of y ¼ 7: To summarize, maximum likelihood estimation is a method to seek the probability distribution that
makes the observed data most likely.
3.1. Likelihood equation
MLE estimates need not exist nor be unique. In this
section, we show how to compute MLE estimates when
they exist and are unique. For computational convenience, the MLE estimate is obtained by maximizing the
log-likelihood function, ln LðwjyÞ: This is because the
two functions, ln LðwjyÞ and LðwjyÞ; are monotonically
related to each other so the same MLE estimate is
obtained by maximizing either one. Assuming that the
log-likelihood function, ln LðwjyÞ; is differentiable, if
wMLE exists, it must satisfy the following partial
differential equation known as the likelihood equation:
@ln LðwjyÞ
¼0
@wi
ð7Þ
93
at wi ¼ wi;MLE for all i ¼ 1; y; k: This is because the
definition of maximum or minimum of a continuous
differentiable function implies that its first derivatives
vanish at such points.
The likelihood equation represents a necessary condition for the existence of an MLE estimate. An
additional condition must also be satisfied to ensure
that ln LðwjyÞ is a maximum and not a minimum, since
the first derivative cannot reveal this. To be a maximum,
the shape of the log-likelihood function should be
convex (it must represent a peak, not a valley) in the
neighborhood of wMLE : This can be checked by
calculating the second derivatives of the log-likelihoods
and showing whether they are all negative at wi ¼ wi;MLE
for i ¼ 1; y; k;1
@ 2 ln LðwjyÞ
o0:
@w2i
ð8Þ
To illustrate the MLE procedure, let us again consider
the previous one-parameter binomial example given a
fixed value of n: First, by taking the logarithm of the
likelihood function Lðwjn ¼ 10; y ¼ 7Þ in Eq. (6), we
obtain the log-likelihood as
10!
þ 7 ln w þ 3 lnð1 wÞ:ð9Þ
ln Lðw j n ¼ 10; y ¼ 7Þ ¼ ln
7!3!
Next, the first derivative of the log-likelihood is
calculated as
d ln Lðw j n ¼ 10; y ¼ 7Þ 7
3
7 10w
¼ ¼
:
dw
w 1 w wð1 wÞ
ð10Þ
By requiring this equation to be zero, the desired MLE
estimate is obtained as wMLE ¼ 0:7: To make sure that
the solution represents a maximum, not a minimum, the
second derivative of the log-likelihood is calculated and
evaluated at w ¼ wMLE ;
d 2 ln Lðw j n ¼ 10; y ¼ 7Þ
7
3
¼ 2
dw2
w
ð1 wÞ2
¼ 47:62o0
ð11Þ
which is negative, as desired.
In practice, however, it is usually not possible to
obtain an analytic form solution for the MLE estimate,
especially when the model involves many parameters
and its PDF is highly non-linear. In such situations, the
MLE estimate must be sought numerically using nonlinear optimization algorithms. The basic idea of nonlinear optimization is to quickly find optimal parameters
that maximize the log-likelihood. This is done by
1
2
Consider
the
Hessian
matrix
HðwÞ
defined
as
Hij ðwÞ ¼
@ ln LðwÞ
ði; j ¼ 1; y; kÞ: Then a more accurate test of the convexity
@wi @wj
condition requires that the determinant of HðwÞ be negative definite,
that is, z0 Hðw ¼ wMLE Þzo0 for any kx1 real-numbered vector z; where
z0 denotes the transpose of z:
94
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
Fig. 3. A schematic plot of the log-likelihood function for a fictitious one-parameter model. Point B is the global maximum whereas points A and C
are two local maxima. The series of arrows depicts an iterative optimization process.
searching much smaller sub-sets of the multi-dimensional parameter space rather than exhaustively searching the whole parameter space, which becomes
intractable as the number of parameters increases. The
‘‘intelligent’’ search proceeds by trial and error over the
course of a series of iterative steps. Specifically, on each
iteration, by taking into account the results from the
previous iteration, a new set of parameter values is
obtained by adding small changes to the previous
parameters in such a way that the new parameters are
likely to lead to improved performance. Different
optimization algorithms differ in how this updating
routine is conducted. The iterative process, as shown by
a series of arrows in Fig. 3, continues until the
parameters are judged to have converged (i.e., point B
in Fig. 3) on the optimal set of parameters on an
appropriately predefined criterion. Examples of the
stopping criterion include the maximum number of
iterations allowed or the minimum amount of change in
parameter values between two successive iterations.
3.2. Local maxima
It is worth noting that the optimization algorithm
does not necessarily guarantee that a set of parameter
values that uniquely maximizes the log-likelihood will be
found. Finding optimum parameters is essentially a
heuristic process in which the optimization algorithm
tries to improve upon an initial set of parameters that is
supplied by the user. Initial parameter values are chosen
either at random or by guessing. Depending upon the
choice of the initial parameter values, the algorithm
could prematurely stop and return a sub-optimal set of
parameter values. This is called the local maxima
problem. As an example, in Fig. 3 note that although
the starting parameter value at point a2 will lead to the
optimal point B called the global maximum, the starting
parameter value at point a1 will lead to point A, which is
a sub-optimal solution. Similarly, the starting parameter
value at a3 will lead to another sub-optimal solution at
point C.
Unfortunately, there exists no general solution to the
local maximum problem. Instead, a variety of techniques have been developed in an attempt to avoid the
problem, though there is no guarantee of their
effectiveness. For example, one may choose different
starting values over multiple runs of the iteration
procedure and then examine the results to see whether
the same solution is obtained repeatedly. When that
happens, one can conclude with some confidence that a
global maximum has been found.2
2
A stochastic optimization algorithm known as simulated annealing
(Kirkpatrick, Gelatt, & Vecchi, 1983) can overcome the local maxima
problem, at least in theory, though the algorithm may not be a feasible
option in practice as it may take an realistically long time to find the
solution.
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
3.3. Relation to least-squares estimation
Recall that in MLE we seek the parameter values that
are most likely to have produced the data. In LSE, on
the other hand, we seek the parameter values that
provide the most accurate description of the data,
measured in terms of how closely the model fits the
data under the square-loss function. Formally, in LSE,
the sum of squares error (SSE) between observations and
predictions is minimized:
m
X
SSEðwÞ ¼
ðyi prdi ðwÞÞ2 ;
ð12Þ
i¼1
where prdi ðwÞ denotes the model’s prediction for the ith
observation. Note that SSEðwÞ is a function of the
parameter vector w ¼ ðw1 ; y; wk Þ:
As in MLE, finding the parameter values that
minimize SSE generally requires use of a non-linear
optimization algorithm. Minimization of LSE is also
subject to the local minima problem, especially when the
model is non-linear with respect to its parameters. The
choice between the two methods of estimation can have
non-trivial consequences. In general, LSE estimates tend
to differ from MLE estimates, especially for data that
are not normally distributed such as proportion correct
and response time. An implication is that one might
possibly arrive at different conclusions about the same
data set depending upon which method of estimation is
employed in analyzing the data. When this occurs, MLE
should be preferred to LSE, unless the probability
density function is unknown or difficult to obtain in an
easily computable form, for instance, for the diffusion
model of recognition memory (Ratcliff, 1978).3 There is
a situation, however, in which the two methods
intersect. This is when observations are independent of
one another and are normally distributed with a
constant variance. In this case, maximization of the
log-likelihood is equivalent to minimization of SSE, and
therefore, the same parameter values are obtained under
either MLE or LSE.
4. Illustrative example
In this section, I present an application example of
maximum likelihood estimation. To illustrate the
method, I chose forgetting data given the recent surge
of interest in this topic (e.g. Rubin & Wenzel, 1996;
Wickens, 1998; Wixted & Ebbesen, 1991).
Among a half-dozen retention functions that have
been proposed and tested in the past, I provide an
example of MLE for the two functions, power and
exponential. Let w ¼ ðw1; w2 Þ be the parameter vector, t
3
For this model, the PDF is expressed as an infinite sum of
transcendental functions.
95
time, and pðw; tÞ the model’s prediction of the probability of correct recall at time t: The two models are
defined as
power model : pðw; tÞ ¼ w1 tw2 ðw1 ; w2 40Þ;
exponential model : pðw; tÞ ¼ w1 expðw2 tÞ
ð13Þ
ðw1 ; w2 40Þ:
Suppose that data y ¼ ðy1 ; y; ym Þ consists of m
observations in which yi ð0pyi p1Þ represents an observed proportion of correct recall at time ti ði ¼
1; y; mÞ: We are interested in testing the viability of
these models. We do this by fitting each to observed data
and examining its goodness of fit.
Application of MLE requires specification of the PDF
f ðyjwÞ of the data under each model. To do this, first we
note that each observed proportion yi is obtained by
dividing the number of correct responses ðxi Þ by the
total number of independent trials ðnÞ; yi ¼
xi =n ð0pyi p1Þ We then note that each xi is binomially
distributed with probability pðw; tÞ so that the PDFs for
the power model and the exponential model are
obtained as
power : f ðxi j n; wÞ ¼
exponential :
n!
ðn xi Þ!xi !
2 xi
2 nxi
ðw1 tw
Þ ð1 w1 tw
Þ
;
i
i
n!
ð14Þ
f ðxi j n; wÞ ¼
ðn xi Þ!xi !
ðw1 expðw2 ti ÞÞxi
ð1 w1 expðw2 ti ÞÞnxi ;
where xi ¼ 0; 1; y; n; i ¼ 1; y; m:
There are two points to be made regarding the PDFs
in the above equation. First, the probability parameter
of a binomial probability distribution (i.e. w in Eq. (4))
is being modeled. Therefore, the PDF for each model in
Eq. (14) is obtained by simply replacing the probability
parameter w in Eq. (4) with the model equation, pðw; tÞ; in
Eq. (13). Second, note that yi is related to xi by a fixed
scaling constant, 1=n: As such, any statistical conclusion
regarding xi is applicable directly to yi ; except for the scale
transformation. In particular, the PDF for yi ; f ðyi jn; wÞ;
is obtained by simply replacing xi in f ðxi jn; wÞ with nyi :
Now, assuming that xi ’s are statistically independent
of one another, the desired log-likelihood function for
the power model is given by
ln Lðw ¼ ðw1 ; w2 Þjn; xÞ
¼ lnðf ðx1 jn; wÞ f ðx2 j n; wÞ?f ðxm j n; wÞÞ
m
X
ln f ðxi jn; wÞ
¼
i¼1
¼
m
X
2
2
ðxi lnðw1 tw
Þ þ ðn xi Þ lnð1 w1 tw
Þ
i
i
i¼1
þ ln n! lnðn xi Þ! ln xi !Þ:
ð15Þ
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
96
Fig. 4. Modeling forgetting data. Squares represent the data in Murdock (1961). The thick (respectively, thin) curves are best fits by the power
(respectively, exponential) models.
Table 1
Summary fits of Murdock (1961) data for the power and exponential models under the maximum likelihood estimation (MLE) method and the leastsquares estimation (LSE) method.
MLE
Loglik/SSE ðr2 Þ
Parameter w1
Parameter w2
LSE
Power
Exponential
Power
Exponential
313:37 ð0:886Þ
0.953
0.498
305:31 ð0:963Þ
1.070
0.131
0.0540 (0.894)
1.003
0.511
0.0169 (0.967)
1.092
0.141
Note: For each model fitted, the first row shows the maximized log-likelihood value for MLE and the minimized sum of squares error value for LSE.
Each number in the parenthesis is the proportion of variance accounted for (i.e. r2 ) in that case. The second and third rows show MLE and LSE
parameter estimates for each of w1 and w2 : The above results were obtained using Matlab code described in the appendix.
This quantity is to be maximized with respect to the
two parameters, w1 and w2 : It is worth noting that the
last three terms of the final expression in the above
equation (i.e., ln n! lnðn xi Þ! ln xi !) do not depend
upon the parameter vector, thereby do not affecting the
MLE results. Accordingly, these terms can be ignored,
and their values are often omitted in the calculation of
the log-likelihood. Similarly, for the exponential model,
its log-likelihood function can be obtained from Eq. (15)
2
by substituting w1 expðw2 ti Þ for w1 tw
:
i
In illustrating MLE, I used a data set from Murdock
(1961). In this experiment subjects were presented with a
set of words or letters and were asked to recall the items
after six different retention intervals, ðt1 ; y; t6 Þ ¼
ð1; 3; 6; 9; 12; 18Þ in seconds and thus, m ¼ 6: The
proportion recall at each retention interval was calculated based on 100 independent trials (i.e. n ¼ 100) to
yield the observed data ðy1 ; y; y6 Þ ¼ ð0:94; 0:77; 0:40;
0:26; 0:24; 0:16Þ; from which the number of correct
responses, xi ; is obtained as 100yi ; i ¼ 1; y; 6: In
Fig. 4, the proportion recall data are shown as squares.
The curves in Fig. 4 are best fits obtained under MLE.
Table 1 summarizes the MLE results, including fit
measures and parameter estimates, and also include the
LSE results, for comparison. Matlab code used for the
calculations is included in the appendix.
The results in Table 1 indicate that under either
method of estimation, the exponential model fit better
than the power model. That is, for the former, the loglikelihood was larger and the SSE smaller than for the
latter. The same conclusion can be drawn even in terms
of r2 : Also note the appreciable discrepancies in
parameter estimate between MLE and LSE. These
differences are not unexpected and are due to the fact
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
that the proportion data are binomially distributed, not
normally distributed. Further, the constant variance assumption required for the equivalence between MLE and LSE
does not hold for binomial data for which the variance, s2 ¼
npð1 pÞ; depends upon proportion correct p:
97
issues in model selection, the reader is referred elsewhere
(e.g. Linhart & Zucchini, 1986; Myung, Forster, &
Browne, 2000; Pitt, Myung, & Zhang, 2002).
5. Concluding remarks
4.1. MLE interpretation
What does it mean when one model fits the data better
than does a competitor model? It is important not to
jump to the conclusion that the former model does a
better job of capturing the underlying process and
therefore represents a closer approximation to the true
model that generated the data. A good fit is a necessary,
but not a sufficient, condition for such a conclusion. A
superior fit (i.e., higher value of the maximized loglikelihood) merely puts the model in a list of candidate
models for further consideration. This is because a model
can achieve a superior fit to its competitors for reasons
that have nothing to do with the model’s fidelity to the
underlying process. For example, it is well established in
statistics that a complex model with many parameters fits
data better than a simple model with few parameters,
even if it is the latter that generated the data. The central
question is then how one should decide among a set of
competing models. A short answer is that a model should
be selected based on its generalizability, which is defined
as a model’s ability to fit current data but also to predict
future data. For a thorough treatment of this and related
This article provides a tutorial exposition of maximum likelihood estimation. MLE is of fundamental
importance in the theory of inference and is a basis of
many inferential techniques in statistics, unlike LSE,
which is primarily a descriptive tool. In this paper, I
provide a simple, intuitive explanation of the method so
that the reader can have a grasp of some of the basic
principles. I hope the reader will apply the method in his
or her mathematical modeling efforts so a plethora of
widely available MLE-based analyses (e.g. Batchelder &
Crowther, 1997; Van Zandt, 2000) can be performed on
data, thereby extracting as much information and
insight as possible into the underlying mental process
under investigation.
Acknowledgments
This work was supported by research Grant R01
MH57472 from the National Institute of Mental Health.
The author thanks Mark Pitt, Richard Schweickert, and
two anonymous reviewers for valuable comments on
earlier versions of this paper.
Appendix
This appendix presents Matlab code that performs MLE and LSE analyses for the example described in the text.
Matlab Code for MLE
% This is the main program that finds MLE estimates. Given a model, it
% takes sample size (n), time intervals (t) and observed proportion correct
% (y) as inputs. It returns the parameter values that maximize the log% likelihood function
global n t x; % define global variables
opts ¼ optimset (‘DerivativeCheck’,‘off’,’Display’,‘off’,‘TolX’,1e-6,‘TolFun’,1e-6,
‘Diagnostics’,‘off’,‘MaxIter’,200,LargeScale’,‘off’);
% option settings for optimization algorithm
n ¼ 100 ;% number of independent Bernoulli trials (i.e., sample size)
t ¼ ½1 3 6 9 12 18
0 ;% time intervals as a column vector
y ¼ ½:94 :77 :40 :26 :24 :16
0 ;% observed proportion correct as a column vector
x ¼ nn y;% number of correct responses
init w ¼ randð2; 1Þ;% starting parameter values
low w ¼ zerosð2; 1Þ;% parameter lower bounds
up w ¼ 100n onesð2; 1Þ;% parameter upper bounds
while 1,
½w1; lik1; exit1
¼ fmincon (‘power mle’,init w,[],[],[],[],low w,up w,[],opts);
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I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
% optimization for power model that minimizes minus log-likelihood (note that minimization of
minus log-likelihood is equivalent to maximization of log-likelihood)
% w1: MLE parameter estimates
% lik1: maximized log-likelihood value
% exit1: optimization has converged if exit1 40 or not otherwise
½w2; lik2; exit2
¼ FMINCONð‘EXPO MLE’; INIT W;½
; ½
; ½
; ½
; LOW W; UP W; ½
; OPTSÞ;
% optimization for exponential model that minimizes minus log-likelihood
prd1 ¼ w1ð1; 1Þn t:#ð-w1ð2; 1ÞÞ;% best fit prediction by power model
r2ð1; 1Þ ¼ 1-sumððprd1-yÞ:#2Þ=sumððy-meanðyÞÞ:#2Þ; % r#2 for power model
prd2 ¼ w2ð1; 1Þn expð-w2ð2; 1Þn tÞ;% best fit prediction by exponential model
r2ð2; 1Þ ¼ 1-sumððprd2-yÞ:#2Þ=sumððy-meanðyÞÞ:#2Þ; %r#2 for exponential model
if sumðr240Þ ¼¼ 2
break;
else
init w ¼ randð2; 1Þ;
end;
end;
format long;
disp(num2str([w1 w2 r2],5));% display results
disp(num2str([lik1 lik2 exit1 exit2],5));% display results
end % end of the main program
function loglik ¼ power mleðwÞ
% POWER MLE The log-likelihood function of the power model
global n t x;
p ¼ wð1; 1Þn t:#ð-wð2; 1ÞÞ;% power model prediction given parameter
p ¼ p þ ðp ¼¼ zerosð6; 1ÞÞn 1e-5 ðp ¼¼ onesð6; 1ÞÞn 1e-5; % ensure 0opo1
loglik ¼ ð1Þn ðx: * logðpÞ þ ðn-xÞ: * logð1-pÞÞ;
% minus log-likelihood for individual observations
loglik ¼ sumðloglikÞ;% overall minus log-likelihood being minimized
function loglik ¼ expo mleðwÞ
% EXPO MLE The log-likelihood function of the exponential model
global n t x;
p ¼ wð1; 1Þn expð-wð2; 1Þn tÞ;% exponential model prediction
p ¼ p þ ðp ¼¼ zerosð6; 1ÞÞn 1e-5 ðp ¼¼ onesð6; 1ÞÞn 1e-5; % ensure 0opo1
loglik ¼ ð1Þn ðx: * logðpÞ þ ðn-xÞ: * logð1pÞÞ;
% minus log-likelihood for individual observations
loglik ¼ sumðloglikÞ;% overall minus log-likelihood being minimized
Matlab Code for LSE
% This is the main program that finds LSE estimates. Given a model, it
% takes sample size (n), time intervals (t) and observed proportion correct
% (y) as inputs. It returns the parameter values that minimize the sum of
% squares error
global t; % define global variable
opts ¼ optimset(‘DerivativeCheck’,‘off’,‘Display’,‘off’,‘TolX’,1e-6,‘TolFun’,1e-6, ‘Diagnostics’,‘off’,‘MaxIter’,200,‘LargeScale’,‘off’);
% option settings for optimization algorithm
n ¼ 100; % number of independent binomial trials (i.e., sample size)
t ¼ ½1 3 6 9 12 18
0 ;% time intervals as a column vector
I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100
99
y ¼ ½:94 :77 :40 :26 :24 :16
0 ;% observed proportion correct as a column vector
init w ¼ randð2; 1Þ;% starting parameter values
low w ¼ zerosð2; 1Þ;% parameter lower bounds
up w ¼ 100n onesð2; 1Þ;% parameter upper bounds
½w1; sse1; res1; exit1
¼ lsqnonlinð‘power lse0 ; init w; low w; up w; opts; yÞ;
% optimization for power model
% w1: LSE estimates
% sse1: minimized SSE value
% res1: value of the residual at the solution
% exit1: optimization has converged if exit1 40 or not otherwise
½w2; sse2; res2; exit2
¼ lsqnonlinð‘expo lse0 ; init w; low w; up w; opts; yÞ;
% optimization for exponential model
r2ð1; 1Þ ¼ 1-sse1=sumððy-meanðyÞÞ:#2Þ; %
r2ð2; 1Þ ¼ 1-sse2=sumððy-meanðyÞÞ:#2Þ; %
r#2 for power model
r#2 for exponential model
format long;
disp(num2str([w1 w2 r2],5));% display out results
disp(num2str([sse1 sse2 exit1 exi2],5));% display out results
end % end of the main program
function dev ¼ power lseðw; yÞ
% POWER LSE The deviation between observation and prediction of the power
% model
global t;
p ¼ wð1; 1Þn t:#ð-wð2; 1ÞÞ;% power model prediction
dev ¼ p y;
% deviation between prediction and observation, the square of which is
being minimized
function dev ¼ expo lseðw; yÞ
% EXPO LSE The deviation between observation and prediction of the
% exponential model
global t;
p ¼ wð1; 1Þn expðwð2; 1Þn tÞ;% exponential model prediction
dev ¼ p y;
% deviation between prediction and observation, the square of which is
being minimized
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